A theme of recent research has been to construct a monomial ideal with a specific minimal free resolution. For example:
(1) The Nearly Scarf construction of a monomial ideal \(M_{\Delta}\) from a simplicial complex \(\Delta\), from [I. Peeva and M. Velasco, Trans. Am. Math. Soc. 363, 2029–2046 (2011; Zbl 1221.13024)];
(2) Faridi’s alternative Scarf constriction that associates an acyclic complex \(\Delta\) to an ideal that has a resolution supported on \(\Delta\) [S. Faridi, J. Commut. Algebra 6, No. 3, 347–361 (2014; Zbl 1330.13016)];
(3) Fløystad’s construction from [G. Fløystad, J. Commut. Algebra 1, No. 1, 57–89 (2009; Zbl 1184.13037)] that associates a maximal monomial ideal \(M_T\) to every tree \(T\).
The paper under review demonstrates that each of these three constructions is a coordinatization of a finite atomic lattice (Section 4, Lemma 4.2, Theorem 5.7). A coordinatization of a lattice \(P\) is an labeling \(\mathcal{M}\) of some elements of \(P\) with non-trivial monomials in a way such that the ideal \(M_{\mathcal{M}}\) produced by this labeling has an lcm-lattice that is isomorphic to \(P\) itself. The authors suggest that this point of view is valuable, because the lcm lattice encodes data that may not be evident when only looking at the original cell structures.
In addition to proving the construction results, the authors show that every monomial ideal can be obtained via a certain coordinatization of its lcm lattice (Prop 2.3) and provide necessary and sufficient conditions to say when a labeling is a coordinatization (Prop 3.2).